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 A360337 G.f. A(x) satisfies: [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n >= 0. 5
 1, 1, 7, 124, 3446, 125706, 5540958, 282129207, 16148101259, 1020687876920, 70377734170699, 5246775452965364, 420104327765022458, 35937961751407922101, 3270668852260460283730, 315546031669853942486219, 32173855061751806476275665, 3457696770952845858846954590 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Paul D. Hanna, Table of n, a(n) for n = 0..300 FORMULA G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies: (1) [x^n] A(x)^(n+1) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) for n>=0. (2) A(x) = Sum_{n>=0} b(n) * x^n/A(x)^n, where b(n) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) / (n+1). a(n) ~ c * d^n * n! * n^alpha, where d = 5.7189630165873859806..., alpha = 1.4141427006501..., c = 0.027880568114272... - Vaclav Kotesovec, Feb 06 2023 EXAMPLE G.f.: A(x) = 1 + x + 7*x^2 + 124*x^3 + 3446*x^4 + 125706*x^5 + 5540958*x^6 + 282129207*x^7 + 16148101259*x^8 + 1020687876920*x^9 + ... RELATED SERIES. G.f. A(x) = B(x/A(x)) where B(x) = B(x*A(x)) begins: B(x) = 1 + x + 8*x^2 + 146*x^3 + 4083*x^4 + 149077*x^5 + 6569555*x^6 + 334401750*x^7 + 19137707066*x^8 + ... + b(n)*x^n + ... such that b(n) = [x^n] (1 + x*A(x)^(3*n+1))^(n+1) / (n+1), as well as b(n) = [x^n] A(x)^(n+1) / (n+1), so that b(n) begin: [1/1, 2/2, 24/3, 584/4, 20415/5, 894462/6, 45986885/7, 2675214000/8, ...]. ILLUSTRATION OF DEFINITION. The table of coefficients of x^k in A(x)^(n+1) begins: n=0: [1, 1, 7, 124, 3446, 125706, 5540958, 282129207, ...]; n=1: [1, 2, 15, 262, 7189, 260040, 11396948, 577954822, ...]; n=2: [1, 3, 24, 415, 11250, 403521, 17583859, 888063051, ...]; n=3: [1, 4, 34, 584, 15651, 556696, 24118370, 1213065672, ...]; n=4: [1, 5, 45, 770, 20415, 720141, 31017985, 1553601145, ...]; n=5: [1, 6, 57, 974, 25566, 894462, 38301069, 1910335764, ...]; n=6: [1, 7, 70, 1197, 31129, 1080296, 45986885, 2283964852, ...]; n=7: [1, 8, 84, 1440, 37130, 1278312, 54095632, 2675214000, ...]; ... Compare to the table of coefficients in (1 + x*A(x)^(3*n+1))^(n+1): n=0: [1, 1, 1, 7, 124, 3446, 125706, 5540958, ...]; n=1: [1, 2, 9, 76, 1252, 32742, 1150522, 49515052, ...]; n=2: [1, 3, 24, 253, 4179, 103866, 3499510, 146421240, ...]; n=3: [1, 4, 46, 584, 10061, 242520, 7836278, 317454824, ...]; n=4: [1, 5, 75, 1115, 20415, 487566, 15193230, 594390940, ...]; n=5: [1, 6, 111, 1892, 37119, 894462, 27139545, 1025356992, ...]; n=6: [1, 7, 154, 2961, 62412, 1538698, 45986885, 1682957396, ...]; n=7: [1, 8, 204, 4368, 98894, 2519232, 75032832, 2675214000, ...]; ... to see that the main diagonals of the tables are the same. PROG (PARI) {a(n) = my(A=[1]); for(m=1, n, A=concat(A, 0); A[m+1] = (Vec((1+x*Ser(A)^(3*m+1))^(m+1))[m+1] - Vec(Ser(A)^(m+1))[m+1])/(m+1) ); A[n+1]} for(n=0, 20, print1(a(n), ", ")) CROSSREFS Cf. A360336, A360338, A360237, A360345, A302703. Sequence in context: A074487 A192566 A322090 * A304420 A217910 A316276 Adjacent sequences: A360334 A360335 A360336 * A360338 A360339 A360340 KEYWORD nonn AUTHOR Paul D. Hanna, Feb 06 2023 STATUS approved

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Last modified September 11 09:25 EDT 2024. Contains 375815 sequences. (Running on oeis4.)